Recall that the bisector of an angle is a ray that separates the angle into two congruent angles.

PS�� bisects �QPR.�QPS � SPRm�QPS � m�SPR

An angle bisector of a triangle is a segment that separates an angle of the triangle into two congruent angles. One of the endpoints of anangle bisector is a vertex of the triangle, and the other endpoint is on the side opposite that vertex.

A�B� is an angle bisector of �DAC.�DAB � �CABm�DAB � m�CAB

Just as every triangle has three medians, three altitudes, and threeperpendicular bisectors, every triangle has three angle bisectors.

An angle bisector of a triangle has all of the characteristics of any anglebisector. In �FGH, F�J� bisects �GFH.

1. �1 � �2, so m�1 � m�2.

2. m�1 � �12

�(m�GFH) or 2(m�1) � m�GFH

3. m�2 � �12

�(m�GFH) or 2(m�2) � m�GFH

H

12

J

F

G

AD

C

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240 Chapter 6 More About Triangles

What You’ll LearnYou’ll learn to identifyand use anglebisectors in triangles.

Why It’s ImportantEngineering Anglebisectors of trianglescan be found inbridges. See Exercise 19.

Angle Bisectors of Triangles6–36–3

Special Segments in Triangles

� altitude � perpendicular � angle bisector

bisector

� line segment � line � ray

� line segment � line segment

from the vertex, a bisects the side of bisects the angleline perpendicular a triangle of a triangleto the opposite side

Segment

Type

Property

Algebra Link

In �MNP, M�O� bisects �NMP.If m�1 � 33, find m�2.

Since M�O� bisects �NMP, m�1 � m�2.

Since m�1 � 33, m�2 � 33.

In �PQR, Q�S� bisects �PQR. If m�PQR � 70, what is m�2?

m�2 � �12

�(m�PQR) Definition of bisector

m�2 � �12

�(70) Substitution

m�2 � 35 Multiply.

In �DEF, E�G� bisects �DEF.If m�1 � 43, find m�DEF.

m�DEF � 2(m�1) Definition of bisectorm�DEF � 2(43) Substitutionm�DEF � 86 Multiply.

In �ABC, A�D� bisects �BAC.

a. If m�1 � 32, find m�2.b. Find m�1 if m�BAC � 52.c. What is m�CAB if m�1 � 28?

In �RST, S�U� is an angle bisector. Find m�UST.

m�UST � m�RSU Definition of bisector5x � 2x � 15 Substitution

5x � 2x � 2x � 15 � 2x Subtract 2x from each side.3x � 15 Simplify.

�33x� � �

135� Divide each side by 3.

x � 5 Simplify.

So, m�UST � 5(5) or 25.

S

U

R

T 5x˚

(2x � 15)˚

A1

2C

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12E

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12

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M1

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O

Lesson 6–3 Angle Bisectors of Triangles 241

Examples

Your Turn

1

2

3

4

Solving Multi-StepEquations, p. 723

Algebra Review

www.geomconcepts.com/extra_examples

Check for UnderstandingCommunicatingMathematics

Guided Practice

Practice

Applications andProblem Solving

1. Describe an angle bisector of a triangle.2. Draw an acute scalene triangle. Then use a

compass and straightedge to construct theangle bisector of one of the angles.

In �DEF, E�G� bisects �DEF, and F�H� bisects �EFD.

3. If m�4 � 24, what is m�DEF?

4. Find m�2 if m�1 � 36.

5. What is m�EFD if m�1 � 42?

6. Algebra In �XYZ, Z�W� bisects �YZX.If m�1 � 5x � 9 and m�2 � 39, find x.

In �ABC, B�D� bisects �ABC, and A�E� bisects �BAC.

7. If m�1 � 55, what is m�ABC?

8. Find m�3 if m�BAC � 38.

9. What is m�4 if m�3 � 22?

10. Find m�2 if m�ABC � 118.

11. What is m�BAC if m�3 � 20?

In �MNP, N�S� bisects �MNP, M�R�bisects �NMP, and P�Q� bisects �MPN.

12. Find m�4 if m�3 � 31.

13. If m�MPN � 34, what is m�6?

14. What is m�3 if m�NMP � 64?

15. Find m�MNP if m�1 � 44.

16. What is m�2 if �MNP is a right angle?

17. In �XYZ, Y�W� bisects �XYZ.What is m�XYZ if m�2 � 62?

18. Algebra In �DEF, E�C� is an angle bisector. If m�CEF � 2x � 10 and m�DEC � x � 25, find m�DEC.

FC

E

D

W

1 2

Z

Y

X

P1

2

3 4

56

S

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N

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43

Z

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XW

1 2

E

D

F12

43

HG

242 Chapter 6 More About Triangles

angle bisector

• • • • • • • • • • • • • • • • • •Exercises

Examples 1–3

Example 4

9, 12 1

8, 10, 13, 14, 16, 20

2

19

7, 11, 15, 17

2, 3

3

18 4

See page 736.Extra Practice

ForExercises

SeeExamples

Homework Help

Mixed Review

19. Engineering One type of bridge, a cable-stayed bridge is shown. Notice that the cable stay anchorage is an angle bisector of each triangle formed by the cables called stays and the roadway. a. Suppose m�ABC � 120, what is m�2?b. Suppose m�4 � 48, what is m�DEF?

20. Critical Thinking What kind of angles are formed when you bisect anobtuse angle of a triangle? Explain.

21. Tell whether the red segment in �ABC is an altitude, a perpendicular bisector, both, or neither. (Lesson 6–2)

22. In �MNP, M�C�, N�B�, and P�A� are medians. Find PD if DA � 6. (Lesson 6–1)

23. Algebra The measures of the angles of a triangle are x � 2, 4x � 3, and x � 7. Find the measure of each angle. (Lesson 5–2)

24. Short Response Triangle DEF has sides that measure 6 feet, 6 feet,and 9 feet. Classify the triangle by its sides. (Lesson 5–1)

25. Multiple Choice Multiply 2r � s by r � 3s. (Algebra Review)2r2 � 3s2 2r2 � 5rs � 3s2 2r2 � 3rs �6r2s2DCBA

C

A B

P

D

N

M

Exercise 22

A

B C

Lesson 6–3 Angle Bisectors of Triangles 243

>

Quiz 1 Lessons 6–1 through 6–3

In �ABC, A�E�, B�F�, and C�D� are medians. (Lesson 6–1)

1. Find GE if AG � 9. 2. What is BF if BG � 5?3. If DG � 12, what is the measure of D�C�?

4. Tell whether P�O� is an altitude, a perpendicular bisector, both, or neither.(Lesson 6–2)

5. Algebra In �DEF, E�G� is an angle bisector. If m�DEG � 2x � 7and m�GEF � 4x � 1, find m�GEF. (Lesson 6–3)

M

O

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D G

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B

A

F

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1 2 3 4

C D F

stays

A

cable stay anchorage

E

Tallmadge Bridge,Savannah, Georgia

Standardized Test Practice

www.geomconcepts.com/self_check_quiz

Materialsruler

protractor

compass

244 Chapter 6 More About Triangles

InvestigationChapter 6

Description of Points Label of Pointsmidpoints of the three sides (3)circumcenter (1)centroid (1)intersection points of altitudes with the sides (3)orthocenter (1)midpoints of segments from orthocenter to each vertex (3)incenter (1)midpoint of segment joining circumcenter and orthocenter (1)

Circumcenter, Centroid,Orthocenter, and IncenterIs there a relationship between the perpendicular bisectors of the sides of a triangle, the medians, the altitudes, and the angle bisectors of atriangle? Let’s find out!

Investigate1. Use construction tools to locate some interesting points on a triangle.

a. Draw a large acute scalene triangle.

b. On a separate sheet of paper, copy the following table.

c. Construct the perpendicular bisector of each side of your triangle.Label the midpoints E, F, and G. Record these letters in your table.The circumcenter is the point where the perpendicular bisectorsmeet. Label this point J and record it. To avoid confusion, erasethe perpendicular bisectors, but not the circumcenter.

d. Draw the medians of your triangle. The point where the mediansmeet is the centroid. Label this point M and record it in your table.Erase the medians, but not the centroid.

Chapter 6 Investigation What a Circle! 245

In this extension, you will determine whether a special circle exists for other types of triangles.

Use paper and construction tools to investigate these cases.

1. an obtuse scalene triangle 2. a right triangle 3. an equilateral triangle

Presenting Your ConclusionsHere are some ideas to help you present your conclusions to the class.

• Make a booklet of your constructions. For each triangle, include a table in which all of the points are recorded.

• Research Leonhard Euler. Write a brief report on his contributions to mathematics,including the nine-point circle.

e. Construct the altitudes of the triangle. Label the points where the altitudes intersect the sides N, P, and Q. Record these points. The point where the altitudes meetis the orthocenter. Label thispoint S and record it. Erase thealtitudes, but not the orthocenter.

f. Draw three segments, each having the orthocenter as one endpointand a vertex of your triangle as the other endpoint. Find the midpointof each segment. Label the midpoints U, V, and W and record thesepoints in your table. Erase the segments.

g. Construct the bisector of each angle of the triangle. The point wherethe angle bisectors meet is the incenter. Label this point X andrecord it. Erase the angle bisectors, but not the incenter.

2. You should now have 13 points labeled. Follow these steps to constructa special circle, called a nine-point circle.

a. Locate the circumcenter and orthocenter. Draw a line segmentconnecting these two points. Bisect this line segment. Label themidpoint Z and record it in the table. Do not erase this segment.

b. Draw a circle whose center is point Z and whose radius extends toa midpoint of the side of your triangle. How many of your labeledpoints lie on or very close to this circle?

c. Extend the segment drawn in Step 2a. This line is called the Euler (OY-ler) line. How many points are on the Euler line?

E

Q G

J

MW V

NP

S

U

F

Investigation For more information on the nine-point circle,visit: www.geomconcepts.com